Quasiconformal distortion barrier for neural warps

First integrate the loss into a neural two-dimensional coordinate map g_theta from the unit square to the image plane. Suitable targets are an image-registration warp, a spatial-transformer sampler, a neural texture-coordinate map, or a planar deformation module. Use g_theta(x) = x + u_theta(x), with u_theta fixed to zero on the boundary or with fixed corner values, so that the learned map remains in the intended homotopy class. At sampled grid points x_i, compute Y_i = g_theta(x_i), the 2 by 2 Jacobian D_i by automatic differentiation or centered finite differences, the determinant J_i = det(D_i), and the singular values of D_i. Set K_i = max(singular_values(D_i))^2 / (J_i + epsilon), compute L_QC = mean(K_i^p), and optimize L_task + lambda_QC L_QC. If J_i is negative, add lambda_fold times softplus(-J_i / tau)^2; do not rely only on determinant clamping. Start with p = 1.5 or 2, then increase p by Delta p after the task loss decreases, stopping at p = 4 or 6. Use random grid samples during training and a dense fixed grid for validation. The mathematics directly supplies D_i, J_i, K_i, and the sampled Lp energy; the global fold rate, minimum determinant, and distortion quantiles are empirical diagnostics. Begin with synthetic checkerboard or MNIST registration using a small CNN, comparing task-only training, task plus squared Jacobian deviation, and task plus quasiconformal distortion. Measure registration error, fold percentage, minimum J, median and 95th-percentile K, and equal-FLOP convergence. Success means fewer folds and lower deformation error at similar task loss.